Download Testing CAPM

Testing CAPM
Plan

Up to now: analysis of return predictability
– Main conclusion: need a better risk model explaining
cross-sectional differences in returns

Today: is CAPM beta a sufficient description of
risks?
– Time-series tests
– Cross-sectional tests
– Anomalies and their interpretation
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CAPM
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Sharpe-Lintner CAPM:
Et-1[Ri,t] = RF + βi (Et-1[RM,t] – RF)
Black (zero-beta) CAPM:
Et-1[Ri,t] = Et-1[RZ,t] + βi (Et-1[RM,t] – Et-1[RZ,t])
Single-period model for expected returns, implying that
– The intercept is zero
– Beta fully captures cross-sectional variation in expected returns

Testing CAPM = checking that market portfolio is on the
mean-variance frontier
– ‘Mean-variance efficiency’ tests
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Testing CAPM

Standard assumptions for testing CAPM
– Rational expectations for Ri,t, RM,t, RZ,t:
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Ex ante → ex post
E.g., Ri,t = Et-1Ri,t + ei,t, where e is white noise
– Constant beta

Testable equations:
Ri,t-RF = βi(RM,t-RF) + εi,t,
Ri,t = (1-βi)RZ,t + βiRM,t + εi,t,
– where Et-1(εi,t)=0, Et-1(RM,tεi,t)=0, Et-1(RZ,tεi,t)=0,
Et-1(εi,t, εi,t+j)=0 (j≠0)
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Time-series tests
Sharpe-Lintner CAPM:
Ri,t-RF = αi + βi(RM,t-RF) + εi,t (+ δiXi,t-1)

– H0: αi=0 for any i=1,…,N (δi=0)

Strong assumptions: Ri,t ~ IID Normal
– Estimate by ML, same as OLS

Finite-sample F-test, which can be rewritten in terms of
Sharpe ratios
– Alternatively: Wald test or LR test

Weaker assumptions: allow non-normality,
heteroscedasticity, auto-correlation of returns
– Test by GMM
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Time-series tests (cont.)
Black (zero-beta) CAPM:
Ri,t = αi + βiRM,t + εi,t,

– H0: there exists γ s.t. αi=(1-βi)γ for any i=1,…,N
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Strong assumptions: Ri,t ~ IID Normal
– LR test with finite-sample adjustment
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Performance of tests:
– The size is correct after the finite-sample adjustment
– The power is fine for small N relative to T
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Results
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Early tests: did not reject CAPM
Gibbons, Ross, and Shanken (1989)
– Data: US, 1926-1982, monthly returns of 11 industry
portfolios, VW-CRSP market index
– For each individual portfolio, standard CAPM is not
rejected
– Joint test rejects CAPM

CLM, Table 5.3
– Data: US, 1965-1994, monthly returns of 10 size
portfolios, VW-CRSP market index
– Joint test rejects CAPM, esp. in the earlier part of the
sample period
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Cross-sectional tests
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Main idea:
Ri,t = γ0 + γ1βi + εi,t (+γ2Xi,t)
H0: asset returns lie on the security market line
– γ0 = RF,
– γ1 = mean(RM-RF) > 0,
– γ2 = 0

Two-stage procedure (Fama-MacBeth, 1973):
– Time-series regressions to estimate beta
– Cross-sectional regressions period-by-period
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Time-series regressions
Ri,t = αi + βiRM,t + εi,t
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First 5y period:
– Estimate betas for individual stocks, form 20 betasorted portfolios with equal number of stocks
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Second 5y period:
– Recalculate betas of the stocks, assign average stock
betas to the portfolios
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Third 5y period:
– Each month, run cross-sectional regressions
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Cross-sectional
regressions
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Ri,t-RF = γ0 + γ1βi + γ2β2i + γ3σi + εi,t
Running this regression for each month t, one gets
the time series of coefficients γ0,t, γ1,t, …
Compute mean and std of γ’s from these time
series:
– No need for s.e. of coefficients in the cross-sectional
regressions!
– Shanken’s correction for the ‘error-in-variables’ problem
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Assuming normal IID returns, t-test
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Why is Fama-MacBeth
approach popular in finance?
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Period-by-period cross-sectional regressions instead of one
panel regression
– The time series of coefficients => can estimate the mean value of
the coefficient and its s.e. over the full period or subperiods
– If coefficients are constant over time, this is equivalent to FE
panel regression
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Simple:
– Avoids estimation of s.e. in the cross-sectional regressions
– Esp. valuable in presence of cross-correlation
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Flexible:
– Easy to accommodate additional regressors
– Easy to generalize to Black CAPM
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Results
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Until late 1970s: CAPM is not rejected
– But: betas are unstable over time
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Since late 70s: multiple anomalies, “fishing license”
on CAPM
– Standard Fama-MacBeth procedure for a given stock
characteristic X:
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Estimate betas of portfolios of stocks sorted by X
Cross-sectional regressions of the ptf excess returns on
estimated betas and X
Reinganum (1981):
– No relation between betas and average returns for betasorted portfolios in 1964-1979 in the US
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Asset pricing anomalies
Variable
Premium's
sign
Reinganum (1983)
January dummy
+
French (1980)
Monday dummy
-
E/P
+
Book-to-market: BE/ME
+
Size: ME
-
Leverage: D/E
+
Jegadeesh & Titman (1993)
Momentum: 6m-1y return
+
De Bondt & Thaler (1985)
Contrarian: 3y-5y return
-
Brennan et al. (1996)
Liquidity: trading volume
-
Basu (1977, 1983)
Stattman (1980)
Banz (1981)
Bhandari (1988)
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Interpretation of
anomalies
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Technical explanations
– There are no real anomalies
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Multiple risk factors
– Anomalous variables proxy additional risk
factors
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Irrational investor behavior
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Technical explanations:
Roll’s critique
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For any ex post MVE portfolio, pricing equations suffice
automatically
It is impossible to test CAPM, since any market index is not
complete
Response to Roll’s critique
– Stambaugh (1982): similar results if add to stock index bonds and
real estate: unable to reject zero-beta CAPM
– Shanken (1987): if correlation between stock index and true global
index exceeds 0.7-0.8, CAPM is rejected

Counter-argument:
– Roll and Ross (1994): even when stock market index is not far
from the frontier, CAPM can be rejected
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Technical explanations:
Data snooping bias

Only the successful results (out of many investigated
variables) are published
– Subsequent studies using variables correlated with those that were
found significant before are also likely to reject CAPM
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Out-of-sample evidence:
– Post-publication performance in US: premiums get smaller (size,
turn of the year effects) or disappear (the week-end, dividend yield
effects)
– Pre-1963 performance in US (Davis, Fama, and French, 2001):
similar value premium, which subsumes the size effect
– Other countries (Fama&French, 1998): value premium in 13
developed countries
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Technical explanations
(cont.)

Error-in-variables problem:
– Betas are measured imprecisely
– Anomalous variables are correlated with true betas
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Sample selection problem
– Survivor bias: the smallest stocks with low returns are excluded
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Sensitivity to the data frequency:
– CAPM not rejected with annual data

Mechanical relation between prices and returns (Berk,
1995)
– Purely random cross-variation in the current prices (Pt)
automatically implies higher returns (Rt=Pt+1/Pt) for low-price
stocks and vice versa
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Multiple risk factors
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Some anomalies are correlated with each
other:
– E.g., size and January effects
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Ball (1978):
– The value effect indicates a fault in CAPM
rather than market inefficiency, since the value
characteristics are stable and easy to observe
=> low info costs and turnover
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Multiple risk factors
(cont.)

Chan and Chen (1991):
– Small firms bear a higher risk of distress, since they are
more sensitive to macroeconomic changes and are less
likely to survive adverse economic conditions
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Lewellen (2002):
– The momentum effect exists for large diversified
portfolios of stocks sorted by size and BE/ME =>
can’t be explained by behavioral biases in info
processing
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Irrational investor
behavior
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Investors overreact to bad earnings =>
temporary undervaluation of value firms
La Porta et al. (1987):
– The size premium is the highest after bad
earnings announcements
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Testing CAPM:
is beta dead ?
Quiz
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Why is it important to have evidence based
on non-US data?
Is Fama-MacBeth approach useful in other
areas than testing CAPM?
What is Roll and Ross (1994) argument ?
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Plan
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Joint test of anomalies
Conditional tests of CAPM
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Fama and French (1992)
"The cross-section of expected stock
returns", a.k.a. "Beta is dead“ article
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Evaluate joint roles of market beta,
size, E/P, leverage, and BE/ME in
explaining cross-sectional variation in
US stock returns
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Data
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All non-financial firms in NYSE, AMEX, and
(after 1972) NASDAQ in 1963-1990
Monthly return data (CRSP)
Annual financial statement data (COMPUSTAT)
– Used with a 6m gap
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Market index: the CRSP value-wtd portfolio of
stocks in the three exchanges
– Alternatively: EW and VW portfolio of NYSE stocks,
similar results (unreported)
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Data (cont.)

‘Anomaly’ variables:
–
–
–
–
Size: ln(ME)
Book-to-market: ln(BE/ME)
Leverage: ln(A/ME) or ln(A/BE)
Earnings-to-price: E/P dummy (1 if E<0) or
E(+)/P
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E/P is a proxy for future earnings only when E>0
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Methodology
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Each year t, in June:
– Determine the NYSE decile breakpoints for size (ME), divide all
stocks to 10 size portfolios
– Divide each size portfolio into 10 portfolios based on pre-ranking
betas estimated over 60 past months
– Measure post-ranking monthly returns of 100 size-beta EW
portfolios for the next 12 months
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Measure full-period betas of 100 size-beta portfolios
Run Fama-MacBeth (month-by-month) CS regressions of
the individual stock excess returns on betas, size, etc.
– Assign to each stock a post-ranking beta of its portfolio
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Results
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Table 1: characteristics of 100 size-beta portfolios
– Panel A: enough variation in returns, small (but not
high-beta) stocks earn higher returns
– Panel B: enough variation in post-ranking betas, strong
negative correlation (on average, -0.988) between size
and beta; in each size decile, post-ranking betas capture
the ordering of pre-ranking betas
– Panel C: in any size decile, the average size is similar
across beta-sorted portfolios
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Results (cont.)
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Table 2: characteristics of portfolios
sorted by size or by pre-ranking beta
– When sorted by size alone: strong
negative relation between size and
returns, strong positive relation between
betas and returns
– When sorted by betas alone: no clear
relation between betas and returns!
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Results (cont.)
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Table 3: Fama-MacBeth regressions
–
–
–
–
Even when alone, beta fails to explain returns!
Size has reliable negative relation with returns
Book-to-market has even stronger (positive) relation
Market and book leverage have significant, but opposite
effect on returns (+/-)

Since coefficients are close in absolute value, this is just
another manifestation of book-to-market effect!
– Earnings-to-price: U-shape, but the significance is killed
by size and BE/ME
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Authors’ conclusions
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“Beta is dead”: no relation between beta and
average returns in 1963-1990
– Other variables correlated with true betas?
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But: beta fails even when alone
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Though: shouldn’t beta be significant because of high negative
correlation with size?
– Noisy beta estimates?
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But: post-ranking betas have low s.e. (most below 0.05)
But: close correspondence between pre- and post-ranking
betas for the beta-sorted portfolios
But: same results if use 5y pre-ranking or 5y post-ranking
betas
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Authors’ conclusions
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Robustness:
– Similar results in subsamples
– Similar results for NYSE stocks in 1941-1990
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Suggest a new model for average returns,
with size and book-to-market equity
– This combination explains well CS variation in
returns and absorbs other anomalies
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Discussion
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Hard to separate size effects from CAPM
– Size and beta are highly correlated
– Since size is measured precisely, and beta is estimated
with large measurement error, size may well subsume
the role of beta!

Once more, Roll and Ross (1994):
– Even portfolios deviating only slightly (within the
sampling error) from mean-variance efficiency may
produce a flat relation between expected returns and
beta
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Further research
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Conditional CAPM
– The ‘anomaly’ variables may proxy for time-varying
market risk exposures
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Consumption-based CAPM
– The ‘anomaly’ variables may proxy for consumption
betas
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Multifactor models
– The ‘anomaly’ variables may proxy for time-varying risk
exposures to multiple factors
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Ferson and Harvey (1998)
"Fundamental determinants of national
equity market returns: A perspective on
conditional asset pricing"
 Conduct conditional tests of CAPM on the
country level
– Relating the instruments to alpha and beta
– Global vs local instruments
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Data
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Monthly returns on MSCI stock indices of 21 developed
countries, 1970-1993
Risk-free rate: US 30-day T-bill
Common instruments (lagged):
– World market return and dividend yield
– The G10 vs USD FX return
– 30d Eurodollar deposit rate, 90-30d Eurodollar term spread

Local instruments:
– Valuation ratios: E/P, P/CF, P/BV, D/P
– Financial: 60m volatility and 6m momentum
– Macro: GDP per capita and inflation (relative to OECD), longterm interest rate, term spread, credit risks
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Methodology

The return-generating process (rational expectations)
ri,t+1 = Et[ri,t+1] + βi,t (rM,t+1-Et[rM,t+1]) + εi,t+1,
– where Et(εi,t+1) = 0, Et(RM,t+1εi,t+1) = 0,
– ri,t+1 is country i’s excess return in US dollars

Model for conditional expected returns and betas:
Et[ri,t+1] = αi,t + βi,t Et[rM,t+1],
–
–
–
–
where βi,t = β0i + β’1iZt + β’2iAi,t,
αi,t = α0i + α’1iZt + α’2iAi,t
Zt are global (world) instruments
Ai,t are local (country-specific) instruments
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Methodology (cont.)

Estimation by GMM (or OLS):
ri,t+1=(α0i+α’1iZt+α’2iAi,t) + (β0i+β’1iZt+β’2iAi,t) rM,t+1+εi,t+1
– H0: αi = 0

Two-factor model:
– Adding foreign exchange risk
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Results

Table 2: conditional betas, joint tests for the
groups of attributes
– For most countries, betas are time-varying
– The impact of global instruments on betas is subsumed
by local variables
– Most important country-specific instruments:
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
for market betas: E/P, inflation, and long-term interest rate
for FX betas: inflation and credit risks
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Results (cont.)

Table 3: conditional alphas in a two-factor model,
leaving only important instruments for betas
– For most countries, alphas are time-varying
– Panel B, jointly significant variables across the
countries: E/P, P/CF, P/BV, volatility, inflation, longterm interest rate, and term spread
– Panel C, economic significance: typical abnormal return
(in response to 1σ change in X) around 1-2% per
month
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Results (cont.)

Table 5: cross-sectional explanatory power
of lagged attributes
– The raw attributes alone produce low R2
– The explanatory power of attributes as
instruments for risk is much greater than for
mispricing
– Some attributes enter mainly as instruments for
beta (e.g., E/P) or alpha (e.g., momentum)
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Methodology II


Fama-MacBeth approach with conditional alpha and betas
in a two-factor model: for each month,
Estimate time-series regression with 60 prior months using
one attribute at a time
ri,t+1 = (α0i + α1iAi,t) + (β0i + β1iAi,t) rW,t+1 + εi,t+1,
– where rW,t+1 is a vector of the world market return and FX rate

Estimate WLS cross-sectional regression using the fitted
values of alpha and/or betas as well as raw attributes:
ri,t+1 = γ0,t+1 + γ1,t+1ai,t+1 + γ’2,t+1bi,t+1 + γ3,t+1Ai,t + ei,t+1
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Conclusions
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Strong support for the conditional asset pricing
model
Local attributes drive out global information
variables in models of conditional betas
The explanatory power of attributes as
instruments for risk is much greater than for
mispricing
The relation of the attributes to expected returns
and risks is different across countries
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